What does Peter learn from the leadership of Adolf Hitler in Ender's Game?

The answer to your question can be found in chapter 9 ("Locke and Demosthenes") of Ender's Game.  During chapter 9, Valentine and Peter vow to be like Demosthenes and Locke respectively.  Peter wants to have more "moderate" ideas about how to truly win the war.  Further, Valentine will be his voice in writing.  First, let us look at the text to find what Peter says about Hitler:

Everybody thinks Hitler got to power because of his armies, because they were willing to kill, and that's partly true, because in the real world power is always built on the threat of death and dishonor.  But mostly he got to power on words, on the right words at the right time.

This is part of Peter's explanation to Valentine of why he needs her to be his voice in writing.  Peter admits some truths about the power in the "real world" here.  Peter uses Hitler as an example of the following truth:  power is, in fact, built on a threat of both dishonor and death.  Even though this is true, there is a hidden truth about Hitler that Peter wants to bring out for Valentine:  Hitler had the power of words behind him.  This is the power that Peter wants Valentine to have.  Just as Hitler "got to power on words, on the right words at the right time," Peter wants Valentine to put convincing words to Peter's ideas so he, too, can use the "right words at the right time" to convince the masses about strategy in battle.